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ASME GT2003-38180
PARALLEL THERMOELASTICITY OPTIMIZATION OF
3-D SERPENTINE COOLING PASSAGES IN TURBINE BLADES
Keywords: Finite elements, least squares, parallel processing, genetic algorithms, stochastic
optimization, thermoelasticity, turbine cooling
This paper was presented as ASME paper GT2003-38180, ASME Turbo Expo 2003, Atlanta, GA,June 16-19, 2003. Parts of this paper will also be published in the book chapter authored byDennis, B. H., Dulikravich, G. S., Yegorov, I. N. and Yoshimura, S., Parallel Optimization of 3-DTurbine Blade Cooling Passages, Chapter 25 in Evolutionary Design Optimization Methods inAeronautical and Turbomachinery Engineering (eds: G. Degrez, J. Periaux, and M. Sefrioui),John Wiley & Sons.
Brian H. DennisFrontier Simulation Software for
Industrial Science Collaborative ResearchCenter, Institute of Industrial ScienceUniversity of Tokyo, 4-6-1 Komaba,
Meguro-ku, Tokyo 153-8505, JAPAN
Igor N. YegorovIOSO Technology Center
Milashenkova 10-201Moscow 127322
RUSSIA
Helmut SobieczkyDLR German Aerospace
CenterBunsenstrasse 1037073 Goettingen
GERMANY
George S. DulikravichFlorida International University
College of Engineering, Room EC 3474Department of Mechanical and Materials
Eng.10555 West Flagler StreetMiami, Florida 33174, USA
Shinobu YoshimuraInstitute of Environmental Studies
Graduate School of Frontier SciencesUniversity of Tokyo
7-3-1 Hongo, Bunkyo-ku,Tokyo 113-8656, JAPAN
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ABSTRACT
An automatic design algorithm for parametric shape optimization of three-dimensional
cooling passages inside axial gas turbine blades has been developed. Smooth serpentine passage
configurations were considered. The geometry of the blade and the internal serpentine cooling
passages were parameterized using surface patch analytic formulation, which provides very high
degree of flexibility, second order smoothness and a minimum number of parameters. The design
variable set defines the geometry of the turbine blade coolant passage including blade wall
thickness distribution and blade internal strut configurations. A parallel three-dimensional
thermoelasticity finite element analysis (FEA) code from the ADVENTURE project at the
University of Tokyo was used to perform automatic thermal and stress analysis of different blade
configurations. The same code can also analyze nonlinear (large/plastic deformation)
thermoelasticity problems for complex 3-D configurations. Convective boundary conditions were
used for the heat conduction analysis to approximate the presence of internal and external fluid
flow.
The objective of the optimization was to make stresses throughout the blade as uniform as
possible. Constraints were that the maximum temperature and stress at any point in the blade
were less than the maximum allowable values. A robust semi-stochastic constrained optimizer
and a parallel genetic algorithm were used to solve this problem while running on an inexpensive
distributed memory parallel computer.
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NOMENCLATURE
F normalized objective function
G1
, G2
inequality constraints
hB outer (hot surface) heat convection coefficient
hC inner (sold surface) heat convection coefficient
n number of computational grid points within the blade
Rf1andRf2 parameters controlling the roundedness of the u-turn shape
Tallow. allowable temperature
TB hot gas bulk temperature
TC coolant bulk temperature
maximum nodal temperature
maximum principal stress at nodei
yield stress of the blade material
maximum nodal principal stress
blade angle with the disk
INTRODUCTION
maxT
i
yield
max
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With the continuing growth of computing resources available, the attention of design
engineers has been rapidly shifting from the use of repetitive computational analysis, personal
experience, and intuition towards a reliable and economical mathematically based optimization
algorithms. Such algorithms have the potential to produce improved designs over a shorter period
of time. In this paper, the application of optimization to the design of passages for internally
cooled 3-D realistic turbine blades is presented.
Internal cooling schemes of modern turbojet and turbofan engines bleed air from the
compressor and pass this air into the serpentine coolant flow passages within the turbine blades.
The maximum temperature within a turbine blade must be kept below a certain value in order to
maintain blade life limited by creep, oxidation, corrosion, and fatigue. To achieve turbine blade
durability requirements, section-averaged centrifugal stress limitations should be satisfied,
concentrations of thermal stress should be limited in the cold areas to reduce low cycle fatigue,
principal strains should be held below a given level in hot areas to reduce thermo-mechanical
fatigue, and the maximum temperature in the blade metal and coating material must be below
specified limits because of oxidation, corrosion and coating spallation concerns. These objectives
can be obtained by the constrained optimization of the coolant passage shapes inside the turbine
blade at a fixed level of coolant flow rate.
There is a strong interaction among a number of engineering disciplines when studying the
design internally cooled gas turbine blades [1,2,3]. The temperature and the associating stresses
within the blade material are considered in detail. However, the effects of the hot gas flow and
coolant flow will be treated in very approximate way. In the design process explained in this
paper, these individual disciplines will not be solved simultaneously in detail for 3-D designs,
because this approach would take an unacceptably long time, even on a cluster of workstations
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running in parallel. For these pragmatic reasons a more approximate yet computationally
affordable design approach was adopted.
The design method is based on a combination of several algorithmic components. These
components include optimization, finite element analysis, and shape design parameterization. The
optimization and finite element analysis modules can be considered black boxes and can be
applied directly to any specific passage design situation. Shape design parameterization on the
other hand is considered problem-specific and different codes need to be developed for different
design problems.
In order to complete the design process in a reasonable amount of time, a parallel computer
should be employed. Both the finite element analysis and the optimization codes used here were
written to make full use of parallel computing resources.
OPTIMIZATION METHOD
The core of the passage design system is the optimization code. The optimizer directs the
design process by generating new designs based on the performance of previous designs, in an
iterative manner. In general, one wishes to use optimization methods that are robust and efficient.
For optimization on a parallel computer, the optimizer should find a good design in the minimum
possible number of iterations. Such algorithms should also be capable of making full use of large-
scale parallel computers. Since each design analysis is a full 3-D simulation, the total computation
time can be from weeks to months if an efficient and sufficiently parallel algorithm is not used.
Robust optimization algorithms are also desired. The optimization process should not
terminate in a local minimum and it should not terminate if the analysis cannot be completed
occasionally due to, for example, failure to generate a proper grid for a candidate design. For the
particular problem considered in this paper, the method should not require gradients of the
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objective or constraints so that discontinuous objective functions and discrete design variables can
be used (for example, total number of cooling passages which could vary during the
optimization).
The core of the design method is the optimization module. The optimizer directs the design
process by generating new designs based on the performance of previous designs, in an iterative
manner. In general, optimization methods that are robust and efficient are the most desirable. For
optimization on a parallel computer, the optimizer should find a good design in the minimum
possible number of iterations. Such algorithms should also be capable of making full use of large-
scale parallel computers. Since each design analysis is a full 3-D simulation, the total computation
time can be from weeks to months if an efficient and sufficiently parallel algorithm is not used. An
optimization method for turbine design should also be robust. The optimizer should not become
trapped in a local minimum due to a noise generated by the analysis code. It should also not
terminate if the analysis cannot be completed due to, for example, failure to generate a proper grid
for a candidate design.
In our experience, genetic algorithm (GA) variations [4,5,6,7] and response surface methods
based on Indirect Optimization based on Self Organization (IOSO) [8,9] work well for 3-D
turbine coolant passage design optimization.
IOSO Method
The IOSO method is a constrained optimization algorithm based on response surface methods
and evolutionary simulation principles. Each iteration of IOSO consists of two steps. The first step
is creation of an approximation of the objective function(s). Each iteration in this step represents a
decomposition of an initial approximation function into a set of simple approximation functions.
The final response function is a multilevel graph such as the one shown in Figure 1. The second
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step is the optimization of this approximation function. This approach allows for self-corrections
of the structure and the parameters of the response surface approximation. The distinctive feature
of this approach is an extremely low number of trial points to initialize the algorithm (30-50
points for the optimization problems with nearly 100 design variables).
Figure 1: Multilevel approximation function
The obtained response functions are used in the procedures of multilevel optimization with the
adaptive changing of the simulation level within the frameworks of both single and multiple
disciplines of the object analysis. During each iteration of the IOSO, the optimization of the
response function is carried out within the current search area. This step is followed by the direct
call to the mathematical model for the obtained point. The information concerning the behavior of
the objective function nearby the extremum is stored, and the response function is made more
accurate just for this search area. For a basic parallel IOSO algorithm, the following steps are
carried out:
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1. Generate a group of designs based on a design of experiments (DOE) method;
2. Evaluate the designs in parallel with the analysis code;
3. Build initial approximation based on the group of evaluated designs;
4. Use stochastic optimization method to find the minimum of the approximation;
5. Do adaptive selection of current extremum search area;
6. Generate a new set of designs in current extremum search area using DOE;
7. Evaluate the new set of designs in parallel with the analysis code;
8. Update the approximation with newly obtained result;
9. Go to 4, unless a termination criterion is met.
Thus, during each iteration, a series of approximation functions is built for a particular
optimization criterion. These functions differ from each other according to both structure and
definition range. The subsequent optimization of the given approximation functions allows us to
determine a set of vectors of optimized variables, which are used to develop further optimization
criteria on a parallel computer.
Multilevel Parallelism in Optimization
The usual approach to parallel optimization is to run a single analysis on each processor per
optimization iteration. However, a mesh for a geometrically complex design may be large;
sometimes the finite element analysis requires more memory than is available on a single
processor. For this reason, the finite element analysis must be distributed among several
processors.
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Figure 2: Multilevel parallelism in optimization.
If a large number of processors are available, the optimizer can use all of them by running several
simultaneous parallel analyses to evaluate several candidate design configurations. For this
research an optimization communication module was developed using the MPI library [10] that
utilizes this multilevel hierarchy of parallelism. This module can be used with any parallel
optimization method including GA and IOSO algorithms. A graphical depiction of the hierarchy
of parallelism is shown in Figure 2.
DESIGN ANALYSIS
The thermal and thermoelastic analysis is performed by parallel finite element analysis. The
finite element analysis codes and tools for mesh generation, mesh partitioning, and others are
freely available as a part of the ADVENTURE project [11] lead by the University of Tokyo. The
Master Optimization process
Analysis control process
ADVENTURE FEM analysis process
Communication by file
Communication by MPI
Design
Variable
s
Objectiv
e&
Const
raints
Subdomain
boundary conditions
ADVENTURE
Input & Output
Files
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finite element solvers are geared towards large-scale parallel analysis and are well suited to the
efficient analysis of complicated geometries.
For each design, a series of modules is required to turn a given set of design variables into
optimization objective and constraint function values. The flow of data between these modules is
depicted graphically in Figure 3. The analysis process may need to be performed hundreds or
thousands of times for a single optimization run so it is critical that each module be automatic,
robust, and computationally efficient.
OBJECTIVE AND CONSTRAINTS
In this section the design objective and constraint functions are detailed. The objective of the
design optimization is to minimize the variation in stress distribution within the blade material.
Figure 3: Modules used for automatic parallel FEA.
Interpreter Geometry
Generator
Mesh
PartitioningApply Boundary
Conditions
Design Variables
Objectives and Constraints
Surface
Mesher
Volume
Mesher
FEA Thermal FEA
ThermoElastic
Compute
Thermal
stresses
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The normalized objective function is computed using the maximum principal stress at each node
within the blade.
(1)
where is the maximum principal stress at nodei,nis the number of computational grid points
within the blade, and is the yield stress of the blade material. Only nodes within the blade
itself are considered for the objective and constraint functions.
By minimizing this objective function, a smoothing effect on the principal stress field is
achieved. In addition, this objective also drives the stresses to lower values, which is also
desirable for the durability of the blade.
In addition to minimizing the objective function, the optimizer must find a design that
simultaneously satisfies the design constraints. For the design of a turbine rotor blade, the
maximum temperature should be less than an allowable temperature, Tallow. Similarly, the
maximum principal stress should be less than the yield stress, . These two inequality
constraints are expressed mathematically as
(2)
=
=n
i yield
i
nF
1
2
i
yield
yield
2
1
1
)(0.100
1
=
= allow
allowin
i T
TT
nG
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(3)
where the constraints are satisfied if and , while
(4)
(5)
The above constraints on maximum temperature and maximum stress could have been written
more simply as
(6)
(7)
where and are the maximum nodal temperature and principal stress, respectively.
However, constraints (2)-(3) have the effect of penalizing designs with many nodes with
2
1
2
)(0.100
1
=
= yield
yieldin
i nG
0.01 G 0.02 G
allowi
allowi
allow
ii
TT
TTif
T
TT
>
=
yieldi
yieldi
yield
ii if
>
=
allowTTG = max1
yieldG = max2
maxT max
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infeasible temperature or stress, where as constraints (6)-(7) only consider the worst values at a
single node. In our experience we found that the constraints (6)-(7) worked well only when an
initial feasible design was given at the start of the optimization. In cases where no initial feasible
design was known, the constraints (2)-(3) produced superior results in fewer iterations for both
GA and IOSO algorithms.
DESIGN PARAMETERIZATION
The outer blade shape was considered to be fixed and to be provided by the user at the
beginning of the design optimization. The shapes of the internal coolant passages were
parameterized using analytical shape functions [12,13]. The turbine blades considered in this
research had a total of four straight passages connected by U-turn passages. The result is a single
serpentine passage with a single inlet and outlet. The spanwise cross-sectional shape of each
straight passage is described by four parameters as shown in Figure 4. These parameters include
the degree of filleting in the passage,r, the blade wall thickness,d, and the passage chordwise
starting and finishing point, x1 and x2 respectively. The passage cross-section shapes are
determined at the root and the tip by user provided parameter values. The parameters for the
middle sections are found by linear interpolation along the blade span.
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Figure 4: Parameters for passage cross-section shape inx-yplane
Figure 5: Parameters for U-turn shape in thex-zplane
Three U-turn shapes are used to connect the ends of the coolant passages. The wall shape of the
U-turn passage is determined by using analytic functions. For walln, the half shape can be found
by using the following equations
(8)
Z1
Z2
Rf2
Rf1
x
z
(xc,zc)
c
R
nn
c
R
cn
zZz
xxxx
fn
fn
n
+=
+=
)sin(
)cos()( max
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wherexmaxis the x-position of the end of the straight passage wall andxc,zcare thex andz
coordinates of the strut center. Four parameters are needed to define each U-turn shape in thex-z
plane as shown in Figure 5. The parametersZ1andZ2control the position of the passage walls in
the z-direction. The parametersRf1andRf2control the roundedness of the u-turn shape. More
details on construction of the turbine blade passages and outer shape are discussed in the
references [12,13].
The straight passage parameterization is somewhat limited because it cannot create designs
with angled struts in thex-yplane. Also in the current approach, the number of straight passages
cannot be changed easily and is fixed at four. These limitations should be addressed to further
increase the usefulness of this approach for creating passage shapes.
Rf1= 0.95,Rf2=0.95
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Rf1= 0.15,Rf2=0.15
Figure 6: Example passage shapes for variation parametersRf1andRf2
The following additional design parameters were also used: the coolant passage bulk temperature,
Tc,and blade angle with the disk,b. All together a total of 42 continuous design variables were
used to uniquely describe a design.
The shape parameterization code generates a block-structured grid that describes the shape of
the blade (Figure 6).
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Figure 7: Internally cooled blade example.
Figure 8: Triangular surface mesh for blade example.
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An inner shroud and blade root geometry are generated separately and added to the base of the
blade section. The block-structured grids for blade, shroud, and root are then used as the base
geometry for generating a triangular surface grid [14]. Sample geometry and the generated
surface mesh are shown in Figures 7-8. The triangular surface mesh is then used as input to a
tetrahedral mesh generation program [15].
DESIGN OPTIMIZATION EXAMPLE
The design system described in this paper was used to perform an example of design optimization
of an internally cooled turbine blade. The outer blade geometry was created by generating a series
of two-dimensional turbine airfoils [6] and stacking the sections along the z-axis. Though the
generated geometry is not an actual turbine blade, we tried to make a simplified outer surface that
maintains the characteristic shape of a typical turbine rotor. However, if a real outer shape is
available from the user, it should be possible to use it directly with the design system with minor
modifications. In this example, the blade material was assumed to be a titanium-aluminum alloy
with the properties listed in Table 1.
Table 1: Physical parameters for Titanium-Aluminum alloy
Modulus of elasticity 118.0 GPaPoisson ratio 0.3
Tensile yield stress,1050.0 MPa
Coefficient of thermal expansion 7.7m m-1 oC-1
Density 4507.0 kg m-3
Thermal conductivity 7.0 W m-1 oC-1
Melting point 1705.0oC
yield
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For each design mesh, the boundary conditions were applied automatically. The root section of
the geometry was set to zero displacement while the blade and inner shroud were left free to
deform. In this simplified problem the aerodynamic loads are not included. As for thermal
boundary conditions, the outer surface of the blade and top surface of inner shroud were set to
convection boundary conditions which require the specification of the convection coefficient, hB,
and the hot gas bulk temperature, TB. Convection boundary conditions were also applied to the
coolant passage surface inside the blade using hCandTC. All other surfaces were assumed
thermally insulated. Both centrifugal and thermal body forces were applied automatically to each
design mesh. Actual values used for this design example are shown in Table 2.
Table 2: Parameters for rotor design problem
The optimization run was performed on a commodity component based PC cluster with 54
Pentium II 400 MHz processors. Both PGA and IOSO optimization methods were tested with this
problem. A total of 12 analyses were performed per iteration for IOSO method. For PGA, 36
designs were evaluated per generation. For both cases each parallel thermoelastic FEM analysis
Coolant convection coefficient,hC 500.0 W m-2 oC-1
Coolant bulk temperature,TC 150.0-600.0oCHot gas convection coefficient,hB 150.0 W m-2 oC-1
Hot gas bulk temperature,TB 1500.0oC
Maximum allowable temperature,Tallow 900.0oCAngular velocity aboutx-axis 5000.0 r.p.m.
Inner shroud distance fromx-axis 0.25 m
Blade span 0.10 m
Blade chord 0.10 m
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used 4 processors. A typical analysis mesh contained over 150,000 degrees of freedom and
required 4 minutes to complete a full thermoelasticity analysis. A converged result was found by
the IOSO optimizer in 70 iterations after consuming approximately 12 hours of total computer
time. For PGA, the total computer time was more than 30 hours. The PGA run was terminated
before a converged result was found. The convergence history for the objective function for both
PGA and IOSO is shown in Figure 9. For all designs the stress constraint was satisfied. However,
the initial design violated the temperature constraint so the optimizer had to first determine a
feasible design. The convergence history for the temperature constraint function is shown in
Figure 10. This figure shows that feasible region was found at iteration 12 for IOSO and iteration
68 for PGA. For the IOSO method, after iteration 12 the best design becomes the feasible design,
which is why a spike in Figure 9 occurs at iteration 12. Prior to that, only infeasible designs were
found and the optimizer clearly tried to improve the objective function while searching for the
feasible region. These convergence results clearly show the computational efficiency of the IOSO
approach over the PGA method for this design problem.
Figure 9: Objective function convergence history.
ITERATION
OBJECTIVE
20 40 60 80
0.003
0.004
0.005
0.006
0.007
0.008
0.009
IOSO
PGA
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Figure 10: Temperature constraint function.
The initial and the IOSO optimized passages configurations are shown in Figures 11 and 12. The
wall near the tip corners has become much thinner in an effort to keep the temperature in those
regions below the maximum allowable value. Principal stresses on the surface of the blade with
the initial shape of the coolant passage is shown in Figure 13, while the IOSO optimized coolant
passage offers lower and more uniform stress field (Figure 14). Stress in the root of the blade is
high due to the centrifugal loading and temperature gradients.
Temperature distributions for the initial design and the IOSO optimized design are shown in
Figures 15-18. The temperature patterns on the surface of the blade follow the shapes of the
passage inside the blade. This shows that the passage shape will have a strong impact on the
temperature distribution and hence the thermally induced stresses. The temperature distribution
on the surface of the IOSO optimized blade is considerably lower and smoother compared with
that of the initial design.
ITERATION
TEMPERATURE
CON
STRAINT
20 40 60 8010
-6
10-5
10-4
10-3
10-2
10-1
100
IOSO
PGA
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Figure 11: Passage shape inx-zplane for initial design.
Figure 12: Passage shape inx-zplane for IOSO optimized design.
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Figure 13: Principal stress contours for initial design.
Figure 14: Principal stress contours for IOSO optimized design.
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Figure 15: Temperature contours for initial design on pressure side.
Figure 16: Temperature contours on pressure side for IOSO optimized design.
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Figure 17: Temperature contours on suction side for initial design.
Figure 18: Temperature contours on suction side for IOSO optimized design.
The IOSO optimized design also includes thick walls near the root of the blade that
progressively thin towards the tip of the blade as shown in Figure 12. This is an expected result
since more material is needed at the root to carry the centrifugal loads.
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Table 3 gives a quantitative comparison between the initial and optimized passage designs.
The initial design exceeds the maximum allowable temperature while satisfying the stress
constraint. However, the optimized blade is clearly feasible with respect to the temperature and
stress constraints.
Table 3: Comparison of initial and optimized cooling passage design
The volume of the optimized blade is slightly smaller than the initial design, most likely due to the
thinning of the walls in the tip region of the blade. The optimized designs maximum principal
stress was reduced by 36 percent and its objective function reduced by 62 percent. The optimizer
reduced the coolant temperature design variable,TC, from 600.0oC to 158.0oC. The reduction in
coolant temperature was necessary for the satisfaction of the maximum temperature constraint.
Although the temperature difference between the coolant and outer hot gas increased, the thermal
stresses actually decreased. The most significant thermal stresses appear to be the result of
Quantity Initial Optimized
Maximum
Temperature,Tmax
1333.8oC 894.6oC
Volume 9.6410-4m3 8.4610-4m3
Maximum Principal
Stress,
668.9 MPa 425.1 MPa
Coolant bulk
temperature,TC
600.0oC 158.0oC
Objective function
value,F
6.8010-3Pa 2.5910-3Pa
max
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temperature distribution along the span and chord direction. Therefore the optimizer determined
the wall thickness distribution such that the variation of temperature in the chord and span
direction was reduced.
CONCLUSIONS
A software system for the design of turbine blade coolant passages has been developed using
powerful optimization algorithms and efficient parallel finite element analysis codes. The
automatic parametric shape design of an internal serpentine coolant passage was demonstrated.
The entire design was completed within 12 hours of computer processing time on a slow small
cluster of processors. A performance comparison was made between PGA and IOSO optimization
methods. For this design problem, the IOSO approach was found to be more efficient by finding
better designs with fewer function evaluations. This example represents a simplified case as the
effect of the inner and outer fluid mechanics is very approximate. The next step towards a
complete automatic design system should be to utilize a reliable three-dimensional fluid
mechanics analysis codes with conjugate heat transfer analysis capability. However, this would
then increase computational requirements by a factor of ten or more. With the recent availability
of low cost parallel supercomputing based commodity component, a complete multidisciplinary
design system may be proven to be computationally and financially feasible in the very near
future.
AKNOWLEDGEMENTS
This work was performed partly in "Frontier Simulation Software for Industrial Science
(FSIS)" project supported by IT program of Japan Ministry of Education, Culture, Sports,
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Science and Technology (MEXT) and partly in the ADVENTURE project of Research for the
Future (RFTF) program supported by the Japan Society for Promotion of Science (JSPS).
The first four authors are grateful for the partial support provided for this research from the
grant NSF DMS-0073698 administered through the Computational Mathematics program
REFERENCES
1.Martin, T. J. and Dulikravich, G. S., 2001, Aero-Thermo-Elastic Concurrent Design
Optimization of Internally Cooled Turbine Blades, Chapter 5 inCoupled Field Problems,
Series on Advances in Boundary Elements(eds: Kassab, A. J. and Aliabadi, M. H.), WIT
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